How exactly do I learn it? First of all, there is this big and expensive Flash with graphical design environment, but I don't use it. I use pure ActionScript 3 programming language (the version is very important here) and FlashDevelop - a great IDE for Windows that looks fast and lightweight but is very convenient and powerful at the same time (I wish IntelliSense in Visual C++ worked like this).
I've been looking for a tutorial to learn the language, but I haven't found anything that would satisfy me, so I've downloaded official documentation from Flash CS3 resources. Until now I've read "Learning ACTIONSCRIPT 3.0" (there is a PDF version for download) and of course I always have the Language Reference opened (there is a downloadable version ActionScriptLangRefV3.zip as ZIP with multipage HTML inside).
Here is my first application - Conway's Game of Life. Use left mouse button to interact.